I was under the same impression as most of you here are, that a person on a visit visa doesn't require an exit permit. But I came to know about this when my brother who came on visit last year was going back after staying here for around 4.5 months. He was asked by the immigration officials at HIA for an exit permit. It came as a total surprise for him and by that time I already had seen off him and reached my home. Then he immediately called me and I rushed to the airport and being a sponsor for him got the exit permit for him from MOI counter at HIA.
So, as I mentioned in my previous post as well, it depends upon the stay duration of the person visiting Qatar.